Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Global Estimates for Singular Integrals of the Composition of the Maximal Operator and the Green’s Operator | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 723234 12 pages doi 2010 723234 Research Article Global Estimates for Singular Integrals of the Composition of the Maximal Operator and the Green s Operator Yi Ling and Hanson M. Umoh Department of Mathematical Sciences Delaware State University Dover DE 19901 USA Correspondence should be addressed to Yi Ling lingyi2001@ and Hanson M. Umoh humoh@ Received 31 December 2009 Accepted 12 March 2010 Academic Editor Shusen Ding Copyright 2010 Y. Ling and H. M. Umoh. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. We establish the Poincare type inequalities for the composition of the maximal operator and the Green s operator in John domains. 1. Introduction Let Q be a bounded convex domain and B a ball in Rn n 2. We use ơB to denote the ball with the same center as B and with diam ơB ơ diam B Ơ 0. We do not distinguish the balls from cubes in this paper. We use E to denote the n-dimensional Lebesgue measure of the set E Q Rn. We say that w is a weight if w e L1oc Rn and w 0 . Differential forms are extensions of functions in Rn. For example the function u x1 x2 . . xn is called a 0-form. Moreover if u x1 x2 . . xn is differentiable then it is called a differential 0-form. The 1-form u x in Rn can be written as u x XZ1 Ui x1 x2 . xn dxi. If the coefficient functions ui x1 x2 . . xn i 1 2 . n are differentiable then u x is called a differential 1-form. Similarly a differential fc-form u x is generated by dxi1 A dxi2 A A dxifc k 1 2 . n that is u x y ui x dxi y ui1i2. ifc x dxi1 A dxi2 A A dxifc I where A is the Wedge Product I Ú i2 . ifc 1 i1 i2 ifc n. Let Al Al Rn 2 Journal of Inequalities and Applications be the set of all -forms in R D q az the space of all differential -forms
